Standards

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Understand and apply theorems about circles
G-C.1. Prove that all circles are similar.
  • I can prove that all circles are similar by showing that for a dilation centered at the center of a circle, the pre-image and the image have equal central angle measures.
  • I can prove that all circles are similar by showing that for a dilation centered at the center of a circle, the pre-image and the image have equal central angle measures.
  • I can recognize the parts of a circle.
  • I can relate the parts of a circle.
  • I can set up ratios for similar circles.
  • I can calculate the circumference of a circle, given the diameter or radius.
  • I can calculate angles inside and outside of a circle.
  • I can compare the ratios of the radius and circumference of multiple circles to determine similarity.
G-C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
  • I can define central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.
  • I can describe the relationship between a central angle and the arc it intercepts.
  • I can describe the relationship between an inscribed angle and the arc it intercepts.
  • I can describe the relationship between a circumscribed angle and the arc it intercepts.
  • I can recognize that an inscribed angle whose sides intersect the endpoints of the diameter of a circle is a right angle.
  • I can recognize that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
  • I can construct and explain examples of central angles, inscribed angles, circumscribed angles, tangent line and chords on a circle.
G-C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
  • I can define the terms inscribed, circumscribed, angle bisector, and perpendicular bisector.
  • I can construct the inscribed circle whose center is the point of intersection of the angle bisectors (the incenter).
  • I can construct the circumscribed circle whose center is the point of intersection of the perpendicular bisectors of each side of the triangle (the circumcenter).
  • I can apply the Arc Addition Postulate to solve for missing arc measures.
  • I can prove that opposite angles in an inscribed quadrilateral are supplementary.
G-C.4. (+) Construct a tangent line from a point outside a given circle to the circle.
  • I can define and identify a tangent line.
  • I can construct a tangent line from a point outside the circle to the circle using construction tools or computer software.

Find arc lengths and areas of sectors of circles

G-C.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
  • I can define similarity as rigid motions with dilations, which preserves angle measures and makes lengths proportional.
  • I can use similarity to calculate the length of an arc.
  • I can define the radian measure of an angle as the ratio of an arc length to its radius and calculate a radian measure when given an arc length and its radius.
  • I can convert degrees to radians using the constant of proportionality 2angle measure360
  • I can calculate the area of a circle.
  • I can define a sector of a circle.
  • I can calculate the area of a sector by deriving the ratio of the intercepted arc and 360⁰ multiplied by the area of the circle.

Experiment with transformations in the plane

G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
  • I can identify undefined notions used in geometry (point, line, plane, distance) and describe their characteristics.
  • I can identify & define angles, circles, perpendicular lines, parallel lines, rays, and line segments.
  • I can define angles, circles, perpendicular lines, parallel lines, rays, and line segments precisely using the undefined terms and “if-then” and “if-and-only-if” statements.
  • I can determine if the lines are parallel, perpendicular or neither based on vertical angles, exterior angles, interior angles, alternate angles.
  • I can use (apply) the symbols for parallel, perpendicular, and angle ( ||,, and <).
  • I can calculate the linear distance and arc length.
9-12.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
  • I can transform a figure on the xy-plane by using different tools (software, graph paper, etc.).
  • I can write a function that will transform a figure on the xy plane.
  • I can determine the coordinates for the image (output) of a figure when a transformation rule is applied to the pre-image (input).
  • I can distinguish between transformations that are rigid (preserve distance and angle measure-reflections, rotations, translations, or combinations of these) and those that are not (dilations or rigid motions followed by dilations).
9-12.G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
  • I can rotate a figure.
  • I can reflect a figure.
  • I can reflect and rotate a figure.
  • I can explain how a figure is reflected onto itself.
  • I can explain how a figure is rotated onto itself.
9-12.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  • I can explain and demonstrate rotations, reflections, and translations.
  • I can use rotations, reflections, and translations to manipulate figures to show congruence.
  • I can explain and demonstrate how transformations are results of manipulations of angles, circles, perpendicular lines, parallel lines, and line segments.
G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
  • I can draw a specific transformation when given a geometric figure and a rotation, reflection, or translation.
  • I can predict and verify the sequence of transformations (a composition) that will map a figure onto another.
  • I can explain why a transformation is an isometry.

Understand congruence in terms of rigid motions

G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
  • I can define rigid motions as reflections, rotations, translations, and combinations of these, all of which preserve distance and angle measure.
  • I can define congruent figures as figures that have the same shape and size and state that a composition of rigid motions will map one congruent figure onto another.
  • I can predict the composition of transformations that will map a figure onto a congruent figure.
  • I can determine if two figures are congruent by determining if rigid motions will turn one figure into the other.
G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
  • I can identify corresponding sides and corresponding angles of congruent triangles.
  • I can explain that in a pair of congruent triangles, corresponding sides are congruent (distance is preserved) and corresponding angles are congruent (angle measure is preserved).
  • I can demonstrate that when distance is preserved (corresponding sides are congruent) and angle measure is preserved (corresponding angles are congruent) the triangles must also be congruent.
  • I can justify congruence of two triangles using isometries.
G-CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
  • I can define rigid motions as reflections, rotations, translations, and combinations of these, all of which preserve distance and angle measure.
  • I can list the sufficient conditions to prove triangles are congruent.
  • I can map a triangle with one of the sufficient conditions (e.g., SSS) onto the original triangle and show that corresponding sides and corresponding angles are congruent.

Prove geometric theorems

G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
  • I can identify and use the properties of congruence and equality (reflexive, symmetric, transitive) in my proofs.
  • I can order statements based on the Law of Syllogism when constructing my proof.
  • I can correctly interpret geometric diagrams by identifying what can and cannot be assumed.
  • I can use the theorems, postulates, or definitions to prove theorems about lines and angles, including:
a) vertical angles are congruent
b) when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and same-side interior angles are supplementary.
c) points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
d) I can prove the converse of the alternate interior angle theorem and the corresponding angle theorem and use it to show that two lines are parallel.
G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
  • I can order statements based on the Law of Syllogism when constructing my proof.
  • I can correctly interpret geometric diagrams (what can and cannot be assumed).
  • I can use theorems, postulates, or definitions to prove theorems about triangles, including
a) measures of interior angles of a triangle sum to 180⁰.
b) base angles of isosceles triangles are congruent.
c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.
d) the medians of a triangle meet at a point.
e) the measure of the exterior angle of a triangle is equal to sum of the two remote interior angles.
G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
  • I can use theorems, postulates, or definitions to prove theorems about parallelograms, including:
    • Prove opposite sides of a parallelogram are congruent
    • Prove opposite angles of a parallelogram are congruent
    • Prove the diagonals of a parallelogram bisect each other
  • I can prove the theorems of special parallelograms.

Make geometric constructions

G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
  • I can identify the tools used in formal constructions.
  • I can use tools and methods to precisely copy a segment, copy an angle, bisect a segment, bisect an angle, construct perpendicular lines and bisectors, and construct a line parallel to a given line through a point not on the line.
  • I can informally perform the constructions listed above using string, reflective devices, paper folding, and/or dynamic geometric software.
G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
  • I can define inscribed polygons (the vertices of the figure must be points on the circle.
  • I can construct an equilateral triangle inscribed in a circle.
  • I can construct a square inscribed in a circle.
  • I can construct a hexagon inscribed in a circle.
  • I can explain the steps to construction an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
  • I can use dynamic geometry software to complete these constructions.

Explain volume formulas and use them to solve problems

G-GMD.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
  • Circumference of a circle
    • I can define (pi) as the ratio of a circle’s circumference to its diameter.
    • I can use algebra to demonstrate that because (pi) is the ratio of a circle’s circumference to its diameter that the formula for a circle’s circumference must be C=d.
  • Area of a circle
    • I can inscribe a regular polygon in a circle and break it into many congruent triangles to find its area.
    • I can explain how to use the dissection method on regular polygons to generate an area formula for regular polygons A=12apothemperimeter.
    • I can calculate the area of a regular polygon.
    • ·I can use pictures to explain that a regular polygon with many sides is nearly a circle, its perimeter is nearly the circumference of a circle, and that its apothem is nearly the radius of a circle.
    • I can substitute the “nearly” values of a many sided regular polygon into A=12apothemperimeterto show that the formula for the area of a circle is A=r2.
  • Volumes
    • I can identify the base for prisms, cylinders, pyramids, and cones.
    • I can calculate the area of the base for prisms, cylinders, pyramids, and cones.
    • I can calculate the volume of a prism using the formula V=Bh and the volume of a cylinder V=r2h.
    • I can defend the statement, “The formula for the volume of a cylinder is basically the same as the formula for the volume of a prism.”
    • I can explain that the volume of a pyramid is 1/3 the volume of a prism with the same base and height and that the volume of a cone is 1/3 the volume of a cylinder with the same base area and height.
    • I can defend the statement, “The formula for the volume of a cone is basically the same as the formula for the volume of a pyramid.”
    • I can decompose volume formulas into area formulas using cross-sections.
    • I can recognize cross-sections for solids as two-dimensional shapes.
    • I can use Cavalieri’s Principle, dissection arguments and informal limit arguments.
G-GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
  • I can calculate the volume of a cylinder and use the volume formula to solve problems.
  • I can calculate the volume of a pyramid and use the volume formula to solve problems.
  • I can calculate the volume of a cone and use the volume formula to solve problems.
  • I can calculate the volume of a sphere and use the volume formula to solve problems.
  • I can identify the solids: cylinders, pyramids, cones, and spheres.

Visualize relationships between two-dimensional and three-dimensional objects
G-GMD.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
  • I can draw a net for a three-dimensional figure.
  • I can identify the shapes of two-dimensional cross-sections of three-dimensional objects (e.g., The cross-section of a sphere is a circle and the cross-section of a rectangular prism is a rectangle, triangle, pentagon, or hexagon.
  • I can rotate a two-dimensional figure and identify the three-dimensional object that is formed (e.g., Rotating a circle produces a sphere, and rotating a rectangle produces a cylinder).
  • I can use geometric simulation software to model figures and create cross-sectional views.
Apply geometric concepts in modeling situations
G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
  • I can represent real-world objects as geometric figures.
  • I can estimate measures (circumference, area, perimeter, volume) of real-world objects using comparable geometric shapes or three-dimensional figures.
  • I can apply the properties of geometric figures to comparable real-world objects (e.g., The spokes of a wheel of a bicycle are equal lengths because they represent the radii of a circle.
  • I can model application problems with geometric shapes.
  • I can find the volume and surface area of a sphere, cylinder, rectangular solid, cone, and pyramid.
G-MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
  • I can decide whether it is best to calculate or estimate the area or volume of a geometric figure and perform the calculation or estimation.
  • I can break composite geometric figures into manageable pieces.
  • I can convert units of measure (e.g., convert square feet to square miles).
  • I can identify the correct labels of density problems, area, and volume.
  • I can build/construct the different volume/area formulas for shapes/figures.
  • I can apply area and volume to situations involving density (e.g., determine the population in an area, the weight of water given its density, or the amount of energy in a three-dimensional figure).
G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
  • I can create a visual representation of a design problem.
  • I can solve design problems using a geometric model (graph, equation, table, formula).
  • I can use simulation software and modeling software to explore which model best describes a set of data or situation.
  • I can interpret the results and make conclusions based on the geometric model.

Translate between the geometric description and the equation for a conic section

G-GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
  • I can identify the center and radius of a circle given its equation.
  • I can draw a right triangle with a horizontal leg, a vertical leg, and the radius of a circle as its hypotenuse.
  • I can use the distance formula (Pythagorean Theorem), the coordinates of a circle’s center, and the circle’s radius to write the equation of a circle.
  • I can convert an equation of a circle in general (quadratic) form to standard form by completing the square.
  • I can identify the center and radius of a circle given its equation.
G-GPE.2. Derive the equation of a parabola given a focus and directrix. (The directrix should be parallel to the coordinate axis.)
  • I can define a parabola.
  • I can find the distance from a point on the parabola (x, y) to the directrix.
  • I can find the distance from a point on the parabola (x, y to the focus using the distance formula (Pythagorean Theorem).
  • I can equate the two distance expressions for a parabola to write its equation.
  • I can identify the focus and directrix of a parabola when given its equation.
  • I can derive the the equation of a parabola given the focus and directrix.

Use coordinates to prove simple geometric theorems algebraically

G-GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
  • I can represent the vertices of a figure in the coordinate plane using variables.
  • I can find the distance and slope between two points on the coordinate plane.
  • I can connect a property of a figure to the tool needed to verify that property.
  • I can use coordinates and the right tool to prove or disprove a claim about a figure.
  • I can use slope to determine if sides are parallel, intersecting, or perpendicular.
  • I can use the distance formula to determine if sides are congruent or to decide if a point is inside a circle, outside a circle, or on the circle.
  • I can use the midpoint formula or the distance formula to decide if a side has been bisected.
  • I can construct formal and informal proofs of geometric theorems.
G-GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
  • Parallel Lines:
    • I can draw a line on a coordinate plane and translate that line to produce its image.
    • I can explain that these lines are parallel since translations preserve angle.
    • I can determine slopes of the original line and its image after translation and show they have the same slope using specific examples and general coordinates (x,y).
    • I can state that parallel line have the same slope.
    • I can determine if lines are parallel using their slope.
    • I can write an equation for a line that is parallel to a given line that passes through a given point.
  • Perpendicular Lines:
    • I can draw a line on a coordinate plane and rotate that line 90⁰ to produce a perpendicular image.
    • I can determine the slope of the original line and its image after a 90⁰ rotation and show they have the opposite reciprocal slopes using specific examples and general coordinates (x, y)
    • I can state that perpendicular lines have the opposite reciprocal slopes.
    • I can determine if lines are perpendicular using their slope.
    • I can write an equation for a line that is perpendicular to a given line that passes through a given point.
G-GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio: I can calculate the point(s) on a directed line segment whose endpoints are x1, y1 and x2, y2that partitions the line segment into a given ratio, r1to r2using the formula x = r2x1 + r1x2r1 + r2, y =r2y1 + r1y2r1 + r2 or [x1+n(x2-x1), y1+n(y2-y1)] where n ≤ 1 (given partition or ratio) (e.g., for the directed line segment whose endpoints are (0,0) and (4,3), the point that partitions the segment into a ratio of 3 to 2 can be found x= 20+343+2=125 and y = 20+ 333+2 = 95 so the point is 125,95 .

G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

  • I can use the coordinates of the vertices of a polygon graphed in the coordinate plane and use the distance formula to compute the perimeter.
  • I can use the coordinates of the vertices of triangles and rectangles graphed in the coordinate plane to compute area.
  • I can use appropriate labels for area & perimeter.

Understand similarity in terms of similarity transformations

G-SRT.1. Verify experimentally the properties of dilations given by a center and a scale factor:
a) A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b) The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
  • I can define dilation.
  • I can perform a dilation with a given center and scale factor on a figure in the coordinate plane.
  • I can verify that when a side passes through the center of dilation, the side and its image lie on the same line.
  • I can verify that corresponding sides of the pre-image and images are proportional.
  • I can verify that a side length of the image is equal to the scale factor multiplied by the corresponding side length of the pre-image.
  • I can use appropriate units of measure when converting scale.
G-SRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
  • I can define similarity as a composition of rigid motions followed by dilations in which angle measure is preserved and side length is proportional.
  • I can identify corresponding sides and corresponding angles of similar triangles.
  • I can demonstrate that in a pair of similar triangles, corresponding angles are congruent (angle measure preserved) and corresponding sides are proportional.
  • I can determine that two figures are similar by verifying that angle measure is preserved and corresponding sides are proportional.
G-SRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
  • I can show and explain that when two angle measures are known (AA), the third angle measure is also known (Third Angle Theorem).
  • I can conclude and explain that AA similarity is a sufficient condition for two triangles to be similar.
  • I can apply similarity transformations to triangles and then verify that the corresponding angles are congruent.

Prove theorems involving similarity

G-SRT.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
  • I can determine if two triangles are similar.
  • I can determine if two lines are parallel.
  • I can write and solve proportions.
  • I can organize & write mathematical proof.
  • I can use theorems, postulates, or definitions to prove theorems about triangles, including
    • A line parallel to one side of a triangle divides the other two proportionally
    • If a line divides two sides of a triangle proportionally, then it is parallel to the third side
    • The Pythagorean Theorem proved using triangle similarity.
G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures
  • I can use triangle congruence and triangle similarity to solve problems (i.e., indirect measure, missing sides/angle measure, side splitting).
  • I can use triangle congruence and triangle similarity to prove relationships in geometric figures.
  • I can justify why two figures are congruent.

Define trigonometric ratios and solve problems involving right triangles

G-SRT.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
  • I can demonstrate that within a right triangle, line segments parallel to a leg create similar triangles by angle-angle similarity (e.g., In triangle ABC where C is the right angle, segment DE can be drawn parallel to segment BC. Since angle A is congruent to angle A and angle AED is congruent to angle ACB, triangle AED is similar to triangle ACB).
  • I can use the characteristics of similar figures to justify trigonometric ratios.
  • I can define trigonometric ratios for acute angles in a right triangle: sine, cosine, tangent, secant, cosecant, cotangent.
  • I can use division and the Pythagorean Theorem c2=a2+b2 to prove that sin2A+cos2A = 1.
G-SRT.7. Explain and use the relationship between the sine and cosine of complementary angles.
  • I can define complementary angles.
  • I can calculate sine and cosine ratios for acute angles in a right triangle when given two side lengths.
  • I can use a diagram of a right triangle to explain that for a pair of complementary angles A and B, the sine of angle A is equal to the cosine of angle B and the cosine of angle A is equal to the sine of angle B.
G-SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
  • I can use angle measures to estimate side lengths (e.g., The side across from a 33⁰ angle will be shorter than the side across from the 57⁰ angle).
  • I can use side lengths to estimate angle measures (e.g., The angle opposite of a 10 cm side will be larger than the angle across from the 9 cm side).
  • I can solve right triangles by finding the measures of all sides and angles in the triangle.
  • I can use sine, cosine, tangent, and their inverses to solve for the unknown side lengths and angle measures of a right triangle.
  • I can draw right triangles that describe real-world problems and label the sides of the angles with their given measures.
  • I can solve application problems involving right triangles, including angle of elevation and depression, navigation, and surveying.

Apply trigonometry to general triangles

G-SRT.9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
  • I can understand that two right triangles are created when an altitude is drawn from a vertex.
  • I can find the length of a triangle’s altitude by using the sine function.
  • I can use the traditional area formula of a triangle A=12baseheight and the sine function to generate an equivalent area formula A=12absinC, using any angle of the triangle.
G-SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
  • I can derive the Law of SInes by drawing an altitude in a triangle, using the sine function to find two expressions for the length of the altitude, and simplifying the equation that results from setting these expressions equal sin Aa =sin Bb = sin Cc.
  • I can use the Law of Sines to solve real world problems.
  • I can use derive the law of cosines.
  • I can use the law of cosines to solve problems.
G-SRT.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
  • I can use the triangle inequality and side/angle relationships (e.g., largest angle is opposite the largest side) to estimate the measures of unknown sides and angles.
  • I can distinguish between situations that require the Law of SInes (ASA, AAS, SSA) and situations that require the Law of Cosines (SAS, SSS).
  • I can apply the Law of Sines to find unknown side lengths and unknown angle measures in right and non-right triangles.
  • I can use the Law of Sines to determine if two given side lengths and a given non-adjacent angle measures (SSA) make two triangles, one triangle, or no triangle.
  • I can apply the Law of Cosines to find unknown side lengths and unknown angle measures in right and non-right triangles.
  • I can represent application problems with diagrams of right and non-right triangles and use them to solve for unknown side lengths and angle measures.

Understand independence and conditional probability and use them to interpret data

S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
  • I can define event and sample space.
  • I can establish events as subsets of a sample space.
  • I can define union, intersection, and complement.
  • I can establish events as subsets of a sample space based on the union, intersection, and/or complement of other events.
  • I can organize outcomes of an event in tree diagrams, Venn diagrams, or contingency tables.
  • I can analyze data and events to determine unions, intersections and/or complements from sample sets.
S-CP.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
  • I can find the probability of two (or more) independent events.
  • I can distinguish between independent and dependent events.
  • I can explain and provide an example to illustrate that for two independent events, the probability of the events occurring together is the product of the probability of each event.
  • I can predict if two events are independent, explain my reasoning, and check my statement by calculating P(A and B) and P(A) x P(B).
S-CP.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
  • I can define dependent events and conditional probability.
  • I can explain that conditional probability is the probability of an event occurring given the occurrence of some other event and give examples that illustrate conditional probability.
  • I can calculate the conditional probability of independent events.
  • I can explain that for two events A and B, the probability of event A occurring given the occurrences of event B is PA|B = PA and BPB and give examples to show how to use the formula.
  • I can explain that A and B are independent events if the occurrences of A does not impact the probability of B occurring and vice versa /i.e., A and B are independent events if P(B│A) = P(B) and P(A│B) = P(A).
  • I can determine if two events are independent and justify my conclusion.
S-CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
  • I can determine when a two way frequency table is an appropriate display for a set of data.
  • I can collect data from a random sample.
  • I can construct a two-way frequency table for the data using the appropriate categories for each variable.
  • I can calculate the conditional probability of A given B using the formula PA|B = PA and BPB.
  • I can decide if events are independent of each other by comparing P(B|A) and P(B) or P(A|B) and P(A) & estimate conditional probability data from the data.
  • I can pose a question for which a two-way frequency is appropriate, use statistical techniques to sample the population, and design an appropriate product to summarize the process and report the results.
  • I can define bivariate data.
S-CP.5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
  • I can illustrate the concept of conditional probability using everyday examples of dependent events.
  • I can illustrate the concept of independence using everyday examples of independent events.

Use the rules of probability to compute probabilities of compound events in a uniform probability model

S-CP.6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
  • I can calculate the probability of the intersection of two events.
  • I can calculate the conditional probability of A given B using the model PA|B = PA and BPB
  • I can interpret probability based on the context of the given problem.
S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
  • I can apply the Addition Rule to determine the probability of the union of two events using the formula P(A or B) = P(A) + P(B) – P(A and B) .
  • I can interpret the probability of unions and intersections based on the context of the given problem.
S-CP.8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
  • I can apply the general Multiplication Rule to calculate the probability of the intersection of two events using the formula P(A and B) = P(A)P(B|A) = P(B)P(A|B)
  • I can interpret conditional probability based on the context of the given problem.
S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
  • I can apply the fundamental counting principle to find the total number of possible outcomes in a sample space.
  • I can define factorial, permutation, combination, and compound event.
  • I can distinguish between situations that require permutations and those that require combinations.
  • I can apply the permutation formula to determine the number of outcomes in an event. nPr = n!n-r! where n = the number of objects to choose from and k = the number of objects selected.
  • I can compute probabilities of compound events.
  • I can solve problems involving permutations and combinations.
  • I can write and solve original problems involving compound events, permutations, and/or combinations.

Use probability to evaluate outcomes of decisions

S-MD.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
  • I can use probability to create a methods with and without technology to make fair decisions.
  • I can use probability to analyze the results of a process and decide if it resulted in a fair decision.
  • I can determine experimental probability of an event and compare it to the theoretical.
S-MD.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
  • I can analyze data to determine whether or not the best decision was made.
  • I can analyze the available strategies, recommend a strategy, and defend my choice.